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H • If x0 is a critical point such that H(x0 ) is positive deﬁnite, then f has a
local minimum at x0 . • If x0 is a critical point such that H(x0 ) is negative deﬁnite (i.e., zT Hz < 0
for all z = 0 or, equivalently, −H is positive deﬁnite), then f has a local
maximum at x0 . IG
R Exercises for section 7.6 Y
P 7.6.1. Which of the following matrices are positive deﬁnite?
⎛
⎞
⎛
⎞
⎛
1 −1 −1
20 6 8
2
A = ⎝ −1
5
1⎠. B = ⎝ 6 3 0⎠. C = ⎝0
−1
1
5
808
2 O
C 0
6
2 ⎞
2
2⎠.
4 7.6.2. Spring-Mass Vibrations. Two masses m1 and m2 are suspended
between three identical springs (with spring constant k ) as shown in
Figure 7.6.7. Each mass is initially displaced from its equilibrium position by a horizontal distance and released to vibrate freely (assume there
is no vertical displacement).
m1 m2 x1 x2 m1 m2 Figure 7.6.7 Copyright c 2000 SIAM Buy online from SIAM
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7.6 Positive Deﬁnite Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540 571 (a) If xi (t) denotes the horizontal displacement of mi from equilibrium at
time t, show that Mx = Kx, where It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu M= m1
0 0
m2 , x= x1 (t)
x2 (t) , and K=k 2
−1 −1
2 . (Consider a force directed to the left to be positive.) Notice that the
mass-stiﬀness equation Mx = Kx is the matrix version of Hooke’s
law F = kx, and K is positive deﬁnite.
(b) Look for a solution of the form x = eiθt v for a constant vector v, and
show that this reduces the problem to solving an algebraic equation of
the form Kv = λMv (for λ = −θ2 ). This is called a generalized
eigenvalue problem because when M = I we are back to the ordinary eigenvalue problem. The generalized eigenvalues λ1 and λ2 are
the roots of the equation det (K − λM) = 0—ﬁnd them when k = 1,
m1 = 1, and m2 = 2, and describe the two modes of vibration.
(c) Take m1 = m2 = m, and...

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